Apply the Tangent Ratio 5.2 (M2)

Vocabulary Trigonometry: branch of mathematics that deals with the relationships between the sides and angles of triangles and the calculations based on these relationships Trigonometric ratio: lengths of 2 sides in a right triangle Tangent of the angle: ratio of the length of the leg opposite an acute angle to the length of the leg adjacent to the angle (constant) C B A

Complementary Angles: sum of their measures is 90 o C B A Can you do the tangent of C?

EXAMPLE 1 Find tangent ratios Find tan S and tan R. Write each answer as a fraction and as a decimal rounded to four places. SOLUTION tan S = opp S adj. to S = RT ST = = tan R = opp R adj. to R = ST RT = = =

GUIDED PRACTICE for Example 1 Find tan J and tan K. Round to four decimal places. ANSWER , ANSWER ,

EXAMPLE 2 Find a leg length ALGEBRA Find the value of x. SOLUTION Use the tangent of an acute angle to find a leg length. tan 32 o = opp. adj. Write ratio for tangent of 32 o. tan 32 o 11 = x Substitute. x tan 32 o = 11 Multiply each side by x. x = 11 tan 32 o Divide each side by tan 32 o x Use a calculator to find tan 32 o x 17.6 Simplify

EXAMPLE 3 Estimate height using tangent LAMPPOST Find the height h of the lamppost to the nearest inch. tan 70 o = opp. adj. Write ratio for tangent of 70 o. tan 70 o h = 40 Substitute. 40 tan 70 o = h Multiply each side by h Use a calculator to simplify. ANSWER The lamppost is about 110 inches tall.

EXAMPLE 4 Use a special right triangle to find a tangent Use a special right triangle to find the tangent of a 60 o angle. STEP 1 Because all 30 o -60 o -90 o triangles are similar, you can simplify your calculations by choosing 1 as the length of the shorter leg. Use the 30 o -60 o -90 o Triangle Theorem to find the length of the longer leg.

EXAMPLE 4 Use a special right triangle to find a tangent longer leg = shorter leg 3 30 o - 60 o - 90 o Triangle Theorem x = 1 3 Substitute. x = 3 Simplify.

EXAMPLE 4 Use a special right triangle to find a tangent STEP 2 tan 60 o = opp. adj. Write ratio for tangent of 60 o. tan 60 o = 3 1 Substitute. tan 60 o = 3 Simplify. ANSWER The tangent of any 60 o angle is Find tan 60 o

GUIDED PRACTICE for Examples 2, 3, and 4 Find the value of x. Round to the nearest tenth. ANSWER 12.2 ANSWER 19.3 ANSWER shorter leg = 5, longer leg =, 35 tan 60 = 5 53 = 3 What If? In Example 4, suppose the side length of the shorter leg is 5 instead of 1. Show that the tangent of 60° is still equal to. 5. 3